Question

The probability that $A$ speaks truth is $\frac{4}{5}$ while this probability for $B$ is $\frac{3}{4}$. The probability that they contradict each other when asked to speak on a fact is:

• A. $\frac{3}{20}$
• B. $\frac{1}{5}$
• C. $\frac{7}{20}$
• D. $\frac{4}{5}$

Answer 1

The correct option is C $\frac{7}{20}$

The probability of speaking truth of $A$ is $P\left(A\right)=\frac{4}{5}$
The probability of not speaking truth of $A$ is $P\left(\overline{A}\right)=1-\frac{4}{5}=\frac{1}{5}$
The probability of speaking truth of $B$ is $P\left(B\right)=\frac{3}{4}$
The probability of not speaking truth of $B$ is $P\left(\overline{B}\right)=\frac{1}{4}$
The probability of they contradict each other
$=P\left(A\right).P\left(\overline{B}\right)+P\left(\overline{A}\right).P\left(B\right)$
$=\frac{4}{5}\times \frac{1}{4}+\frac{1}{5}\times \frac{3}{4}=\frac{1}{5}+\frac{3}{20}=\frac{7}{20}$